.TH std::asinh(std::complex) 3 "2024.06.10" "http://cppreference.com" "C++ Standard Libary"
.SH NAME
std::asinh(std::complex) \- std::asinh(std::complex)

.SH Synopsis
   Defined in header <complex>
   template< class T >                       \fI(since C++11)\fP
   complex<T> asinh( const complex<T>& z );

   Computes complex arc hyperbolic sine of a complex value z with branch cuts outside
   the interval [−i; +i] along the imaginary axis.

.SH Parameters

   z - complex value

.SH Return value

   If no errors occur, the complex arc hyperbolic sine of z is returned, in the range
   of a strip mathematically unbounded along the real axis and in the interval [−iπ/2;
   +iπ/2] along the imaginary axis.

   Error handling and special values

   Errors are reported consistent with math_errhandling.

   If the implementation supports IEEE floating-point arithmetic,

     * std::asinh(std::conj(z)) == std::conj(std::asinh(z))
     * std::asinh(-z) == -std::asinh(z)
     * If z is (+0,+0), the result is (+0,+0)
     * If z is (x,+∞) (for any positive finite x), the result is (+∞,π/2)
     * If z is (x,NaN) (for any finite x), the result is (NaN,NaN) and FE_INVALID may
       be raised
     * If z is (+∞,y) (for any positive finite y), the result is (+∞,+0)
     * If z is (+∞,+∞), the result is (+∞,π/4)
     * If z is (+∞,NaN), the result is (+∞,NaN)
     * If z is (NaN,+0), the result is (NaN,+0)
     * If z is (NaN,y) (for any finite nonzero y), the result is (NaN,NaN) and
       FE_INVALID may be raised
     * If z is (NaN,+∞), the result is (±∞,NaN) (the sign of the real part is
       unspecified)
     * If z is (NaN,NaN), the result is (NaN,NaN)

.SH Notes

   Although the C++ standard names this function "complex arc hyperbolic sine", the
   inverse functions of the hyperbolic functions are the area functions. Their argument
   is the area of a hyperbolic sector, not an arc. The correct name is "complex inverse
   hyperbolic sine", and, less common, "complex area hyperbolic sine".

   Inverse hyperbolic sine is a multivalued function and requires a branch cut on the
   complex plane. The branch cut is conventionally placed at the line segments (-i∞,-i)
   and (i,i∞) of the imaginary axis.

   The mathematical definition of the principal value of the inverse hyperbolic sine is
   asinh z = ln(z +
   √
   1+z2
   ).

   For any z, asinh(z) =

   asin(iz)
   i

   .

.SH Example


// Run this code

 #include <complex>
 #include <iostream>

 int main()
 {
     std::cout << std::fixed;
     std::complex<double> z1(0.0, -2.0);
     std::cout << "asinh" << z1 << " = " << std::asinh(z1) << '\\n';

     std::complex<double> z2(-0.0, -2);
     std::cout << "asinh" << z2 << " (the other side of the cut) = "
               << std::asinh(z2) << '\\n';

     // for any z, asinh(z) = asin(iz) / i
     std::complex<double> z3(1.0, 2.0);
     std::complex<double> i(0.0, 1.0);
     std::cout << "asinh" << z3 << " = " << std::asinh(z3) << '\\n'
               << "asin" << z3 * i << " / i = " << std::asin(z3 * i) / i << '\\n';
 }

.SH Output:

 asinh(0.000000,-2.000000) = (1.316958,-1.570796)
 asinh(-0.000000,-2.000000) (the other side of the cut) = (-1.316958,-1.570796)
 asinh(1.000000,2.000000) = (1.469352,1.063440)
 asin(-2.000000,1.000000) / i = (1.469352,1.063440)

.SH See also

   acosh(std::complex) computes area hyperbolic cosine of a complex number
   \fI(C++11)\fP             (\\({\\small\\operatorname{arcosh}{z}}\\)arcosh(z))
                       \fI(function template)\fP
   atanh(std::complex) computes area hyperbolic tangent of a complex number
   \fI(C++11)\fP             (\\({\\small\\operatorname{artanh}{z}}\\)artanh(z))
                       \fI(function template)\fP
                       computes hyperbolic sine of a complex number
   sinh(std::complex)  (\\({\\small\\sinh{z}}\\)sinh(z))
                       \fI(function template)\fP
   asinh
   asinhf              computes the inverse hyperbolic sine
   asinhl              (\\({\\small\\operatorname{arsinh}{x}}\\)arsinh(x))
   \fI(C++11)\fP             \fI(function)\fP
   \fI(C++11)\fP
   \fI(C++11)\fP
   C documentation for
   casinh
